Help with special relativity problem (off by a factor of 2)

[weston.pace]

Homework Statement

Special Relativity A.P French Problem 1-9

An eccentric billionaire decides to sterilize his 10⁶ liter swimming pool by boiling the water in it. For heating purposes he uses the fusion reaction:

¹H + ³H → ⁴He + radiant energy

Assuming the heating system is 20% efficient, how much does he pay for the tritium (³H) to raise the pool temperature from 20°C to 100°C? It takes 4.2 joules to raise 1g of water through 1°C. Tritium costs about $5 per cm³ of gas at STP.

Homework Equations

E = mc²

The Attempt at a Solution

Total energy required:
$$E_{total} = 4.2 \frac{J}{g°C} * 10^9 g * 80°C = 3.36*10^{11} J$$

Energy Generated per reaction:
$$E_{reaction} = m_{diff} * c^{2}$$
$$E_{reaction} = ((1.0081 + 3.0170) - 4.0039)amu * 1.6603*10^{-27} * (3*10^{8}\frac{m}{s})^{2} = 3.1682 * 10^{-12} J$$

Atoms per cubic centimeter of gas at STP:

$$\frac{atoms}{cm^3}=6.0221*10^{23} * \frac{1}{22.711} * 10^{-3} = 2.6516 * 10^{19}$$

Final Result (1/0.2) is to account for 20% efficiency:
$$cost=5*\frac{1}{0.2} * \frac{E_{total}}{\frac{E_{reaction}}{\frac{atoms}{cm^3}}} =~ 100,000$$

The answer in the book is $50,000, which would imply I'm off by a factor of two. I've looked over this for some time and can't see my mistake. Can anyone help me spot what I've missed?

 

[mfb]

It is "molecules per cubic centimeter", not atoms.

Spoiler

Tritium gas is T₂.

That is a weird setup. He will spend the first billion (at least) for the fusion reactor.

 

[weston.pace]

That is a weird setup. He will spend the first billion (at least) for the fusion reactor.

I think irrationality is a requirement in physics textbooks problems.

Thanks for the help, that would certainly yield the right answer. Is this just something that should have been obvious from a chemistry standpoint? My chemistry is really rusty.

 

[mfb]

Hmm... it is something you should know, but there is a big difference between knowing it and thinking of it.

 

[voko]

Is this just something that should have been obvious from a chemistry standpoint? My chemistry is really rusty.

Chemistry or physics, the thing to remember is that gases are made of molecules, and the Avogadro constant is how many molecules a mole of gas contains. Then you just need to find what a molecule of the given gas is.